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Clarity and Obfuscation in the Use of Mathematics for Economic Reasoning

One of the interesting claims that I noticed was that some people were saying that mathematics, due to its formal nature, provided economists with clarity. This was then typically followed up with appeals to how economics might become a science by increasingly mathematising. This argument seems entirely dubious to anyone who has ever investigated how science functions. But I do not here wish to either discuss whether economics is a science or if scientists working in other fields really do aspire to mathematical clarity rather than creative innovation.

Instead I would like to consider in what sense mathematics can provide economists with clarity in thinking through certain issues and in what sense it might do just the opposite. I think that a good example of mathematics providing clarity is the case of the Keynesian multiplier, which I have discussed on this blog before. The multiplier, when both imports and consumption are taken into account, generally looks something like this:

We can then manipulate this algebraically to get equilibrium income as such:

Here is an example of a mathematical presentation providing clarity. Even without inputting any numbers into the equation we can immediately discern the factors that will generate equilibrium income, Yt*. It will be a component of autonomous consumption, C0, investment, I, government spending, G, and exports, X, minus autonomous imports, M0. It will also be positively multiplied by the consumption multiplier, c, and negatively multiplied by the import multiplier, m.

We know this because the components of income, as just laid out a moment ago, are in the numerator of the equation, while the multipliers are in the denominator. The multipliers are also being subtracted from/added to 1. The larger the denominator, the smaller the numerator and vice versa. So, anything that “subtracts” from the denominator — e.g. the consumption multiplier — will increase the ratio (same directional effect as adding to the numerator), while anything that “adds” to the denominator — e.g. the import multiplier — will decrease the ratio (same directional effect as subtracting from the numerator.

As we can see, even though this piece of algebra looks somewhat mysterious to someone not familiar with it, nevertheless to the trained eye it is actually possible to intuitively interpret it in a very tangible way. This, I think, is why it provides an example of a use of mathematics that at once provides clarity and insight into the underlying relationships.

Compare this presentation, however, with your typical econometric study. Such studies contain innumerable “black boxes” in that the reasoning behind the assumptions made is often entirely unclear. When it is not unclear and is made explicit (Wynne Godley’s forecasts at the Levy Institute are a model of econometric clarity, for example) one quickly sees that such assumptions are entirely arbitrary — often calling into question the entire endeavor.

One thus spends hours attempting to interpret and reconstruct such a study and, all too often, one comes away realising that the assumptions lead one inevitably to interpret the results as being almost entirely arbitrary. Keynes noted this well in his critique of Tinbergen when he wrote:

The labour it involved must have been enormous. The book is full of intelligence, ingenuity and candour; and I leave it with sentiments of respect for the author. But it has been a nightmare to live with, and I fancy that other readers will find the same. (Pp568)

Indeed, one recognises the labour that goes into such studies — especially if you have undertaken one yourself — but at the same time untangling it becomes “a nightmare to live with”. Why? Because such studies do not promote clarity at all. Instead they promote complete and total obscurantism. The mathematical symbols and manipulations become like a dense fog which the reader has to concentrate the depths of their attention and intelligence upon in order to dissipate, only to find that there is often nothing of substance there in any case.

This is not to say that econometrics is entirely useless. As Keynes says in the Tinbergen critique:

This does not mean that economic material may not supply more elementary cases where the method will be fruitful. Take, for instance, Prof. Tinbergen’s third example-namely, the influence on net investment in railway rolling-stock of the rate of increase in traffic, the rate of profit earned by the railways, the price of pig iron and the rate of interest. Here there seems a reasonable prima facie case for expecting that some of the necessary conditions are satisfied. (Pp567-568)

What Keynes is saying is that if we have a number of variables that we can assume to be very closely and immediately related then the econometric method may prove fruitful. I’ve always thought that a nice example of such a paper that did this entirely correctly was Basil Moore’s classic Unpacking the Post-Keynesian Black Box: Bank Lending and the Money Supply where Moore is extremely careful to lay out and justify the causal relationships before he engages in any econometric analysis.

Alas, however, the question as to what is the “correct” manner in which to undertake such a study remains impossibly hard to define. That gives users of the technique who have not bothered (or have not been able) to think through its methodological problems free reign to engage in nonsense. The reason for this is precisely because these mathematical techniques have a tendency, not to clarity at all, but to obscurantism and the moment one gives people a ticket allowing them to engage in obscurantist practices one runs the risk of spiking the proverbial punch.

The same points could be made in a slightly different manner about mathematical models. But the results are clear: while in certain instances mathematics can be used to increase clarity, in others it can be used to engage in obscurantism. The reason why I think that there should be only a limited place for mathematics in economics is because the risks in allowing it a prominent place are too great as it is the usefulness and relevance of economics as a discipline which is at stake.

It is far, far more difficult to engage in obfuscation and magical nonsense when using plain English than it is when using mathematics; not to mention the fact that it is far easier to catch people out. And as a general rule-of-thumb it is probably not unfair to say that as the number of equations grows, the lack of clarity tends to increase and so too do the difficulties in sorting the wheat from the chaff. It is thus the multiplication and proliferation of equations that tends to give rise to nightmares. I think that is what Lawson, Syll and others are getting at when they express skepticism over too heavy a use of mathematics in economics.

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